Doing the Homework
So how do you actually solve homework problems?
It may help to know how the mind solves problems.
You may be used to problems that you can solve at once, and you may
imagine that all problems are solved this way: if you can't do a problem
at once, you cannot do it at all.
Problems in standardized exams are like this: a typical standardized exam
problem is a simple test of one thing.
But there are other kinds of problems, problems that are composed of several
To solve such a problem, you have to
it ("analyze" means literally, to take apart), and that takes time,
energy, and sometimes creativity.
One difficulty is that the text should be a model, and the text contains
such smooth and polished solutions and proofs -- compared to what students
can produce -- that students may wonder if there is something special
about the mathematicians who originally proved the theorems.
But in fact, the mathematicians who originally developed the theorems did
not come up with anything looking like what is in the textbook.
For example ...
First, there are some obscure results that seem to talk about esoteric
things, like the assortment of results about lengths of lines and
centers of mass by Gilles Personne de Roberval, Gregory St. Vincent,
Evangelista Torricelli that we now say, with hindsight, led to the
Fundamental Theorem of the Calculus.
Frequently, it isn't clear at the time where (if anywhere) these results
Then comes the one or two or more mathematicians who come up with
something that we call the Fundamental Theorem.
James Gregory didn't seem to know the significance of his result,
which is similar to that of
Isaac Barrow, who did.
Of course, this theorem seems very strange to us today, since it is
constructed out of classical geometric notions.
Then comes the one or more mathematicians who the popular books will
say made the great accomplishment.
(who was very conscious of his public image and cultivated a reputation
for "genius") developed a system of mathematical results, including
a version of the Fundamental Theorem of the Calculus; Newton, who was
Barrow's student, is generally regarded as the inventer of the calculus.
Frequently, the great accomplishment doesn't work right, or is
unintelligible, or has errors, or, like Newton's Calculus, all of the
Sometimes we need people, like
simply to make sense out of the invention.
In Liebniz's case, he had to figure out what a function is (this
after Newton was integrating and differentiating functions -- or
at least that is what Newton said he was doing when he was dividing
zero by zero(!?)).
Okay, so people understand what the invention is about, but one is not
supposed to divide zero by zero, so now comes the long train of people
who work out ways to finesse the mess
André Marie Ampère,
who came up with one of the major early versions of the Mean Value
to clean up the mess
Augustin Louis Cauchy,
and his limits, although some limit-like things were already around),
and even clean up the mess made by people who cleaned up the mess
Karl Theodor Wilhelm Weierstrass,
whose epsilons and deltas made the notion of the limit more precise
but --- more work ahead, folks! --- less intelligible), which can
lead some people to think that there must be a better way
who used mathematical logic to circumvent all those nasty limits
... but at what cost?).
And of course, then there are the people who write the texts: in the
case of the Calculus, starting with
Jean Le Rond d'Alembert,
and his Cours d'Analyse.
But that is not it, for the first text really ... well, we can and
should do better.
So after two centuries of successive textbook writers figuring out how
to do things better than their competitors, we have the highly staged
productions you can now buy for outrageous prices.
(For more on the fundamental theorem, see
the Thomas Calculus page on the history of the fundamental theorem.)
This is quite typical of mathematics.
For an entire book about how just one formula took a hundred
years to clean up --- assuming it is now cleaned up --- see Proofs
and Refutations by
So when something first appears, it is long and complicated, and we can see
the huge amount of work involved.
Here are two twentieth century examples.
Meanwhile, the research never ends, as mathematicians develop increasingly
powerful versions of the Fundamental Theorem, from Green's Theorem to
Stokes' Theorem to the recent
of Michael Atiyah and Isadore Singer, for which they won the 2004
Abel Prize, the highest award in mathematics.
spent seven years developing his theory of General Relativity.
This meant working on several interconnected complicated problems.
Both of these are relatively new results, and no doubt we will find easier
ways to present and prove them in the century ahead.
So it is true that even mathematicians find mathematics to be hard.
Sometimes it takes a lot of time and effort to solve a problem that looks
Nothing great was ever accomplished without a lot of
spent seven years constructing a proof of Fermat's Last Theorem, a
deceptively simple-looking problem (show that there is no tuple of
four positive integers x, y, z, n,
with n > 2, such that xn + yn
To analyze a complicated or otherwise hard problem, one tries to take it
apart, concentrate on it (or parts of it, or problems similar to it),
return to it (because you probably won't get it all at once), explore it
from a variety of perspectives, etc.
That is what one should do consciously, with the goal of getting the
Unconscious, the great machine in the largely unseen depths of your
mind, to do some work.
For it is the Unconscious that solves hard problems.
To get your Unconscious to solve a problem, you must do the following:
Keep badgering the Unconscious by returning to the problem and spending
time on it.
The Unconscious is somewhat lazy, and gauges how important things are by
the amount of time and energy the Conscious invests on it: the way to
get the Unconscious to work on something is to repeatedly work on it
Work on the correct problem.
There is a difference between
working on a problem and worrying about it.
When working on the problem, one is conscious of the problem, and aspects
and approaches to the problem, and if that is what one concentrates on,
that is what the Unconscious will try to deal with.
But if one spends the time worrying about the problem, obsessing over one's
inability to solve it quickly, wondering what will happen if the problem
doesn't get solved, all that will do is get the Unconscious to think up
dire consequences of not solving the problem --- and thinking up dire
consequences is one of those things that the Unconscious seems to like to
Getting your Unconscious to work for you rather than against you requires
being in the right
state of Consciousness,
and having the right
So work on the problem, but avoid worrying about it.
And don't be discouraged by ideas that the Unconscious turns up that turn out
to be wrong.
One of the great secrets is that most inspirations are useless: useful
inspirations are remembered because they are so hard won, and come
after a long train of flops.
For more, see the page on
Try approaching it from various angles.
There is not necessarily one right way to do it, so think up examples,
find analogies and related problems, try to find different ways of looking
at it, and in general, find additional material for the Unconscious to use.
So here is the procedure.
Most homework problems are straightforard: here is a description of
specifics about doing a mathematics assignment.
Proofs can pose particular difficulties, and there is a page dedicated